In All Life is Problem Solving (1999), Popper sought to explain the apparent progress of scientific knowledge—how it is that our understanding of the universe seems to improve over time. This problem arises from his position that the truth content of our theories, even the best of them, cannot be verified by scientific testing, but can only be falsified. If so, then how is it that the growth of science appears to result in a growth in knowledge? In Popper's view, the advance of scientific knowledge is an evolutionary process characterised by his formula:

In response to a given problem situation (), a number of competing conjectures, or tentative theories (), are systematically subjected to the most rigorous attempts at falsification possible. This process, error elimination (), performs a similar function for science that natural selection performs for biological evolution. Theories that better survive the process of refutation are not more true, but rather, more "fit"—in other words, more applicable to the problem situation at hand (). Consequently, just as a species' "biological fit" does not predict continued survival, neither does rigorous testing protect a scientific theory from refutation in the future. Yet, as it appears that the engine of biological evolution has produced, over time, adaptive traits equipped to deal with more and more complex problems of survival, likewise, the evolution of theories through the scientific method may, in Popper's view, reflect a certain type of progress: toward more and more interesting problems (). For Popper, it is in the interplay between the tentative theories (conjectures) and error elimination (refutation) that scientific knowledge advances toward greater and greater problems; in a process very much akin to the interplay between genetic variation and natural selection.

Where does "truth" fit into all this? As early as 1934 Popper wrote of the search for truth as one of the "strongest motives for scientific discovery." Still, he describes in Objective Knowledge (1972) early concerns about the much-criticised notion of truth as correspondence. Then came the semantic theory of truth formulated by the logician Alfred Tarski. (Its first published form was in 1933.) Popper writes of learning in 1935 of the consequences of Tarski's theory, to his intense joy. The theory met critical objections to truth as correspondence and thereby rehabilitated it. The theory also seemed to Popper to support metaphysical realism and the regulative idea of a search for truth.

According to this theory, the conditions for the truth of a sentence as well as the sentences themselves are part of a metalanguage. So, for example, the sentence "Snow is white" is true if and only if snow is white. Although many philosophers have interpreted, and continue to interpret, Tarski's theory as a deflationary theory, Popper refers to it as a theory in which "is true" is replaced with "corresponds to the facts." He bases this interpretation on the fact that examples such as the one described above refer to two things: assertions and the facts to which they refer. He identifies Tarski's formulation of the truth conditions of sentences as the introduction of a "metalinguistic predicate" and distinguishes the following cases:

1) "John called" is true.
2) "It is true that John called."

The first case belongs to the metalanguage whereas the second is more likely to belong to the object language. Hence, "it is true that" possesses the logical status of a redundancy. "Is true", on the other hand, is a predicate necessary for making general observations such as "John was telling the truth about Phillip."

Upon this basis, along with that of the logical content of assertions (where logical content is inversely proportional to probability), Popper went on to develop his important notion of verisimilitude.
In words, the intuitive idea behind verisimilitude is that the assertions or hypotheses of scientific theories can be objectively measured with respect to the amount of truth and falsity that they imply. And, in this way, one theory can be evaluated as more or less true than another on a quantitative basis which, Popper emphasizes forcefully, has nothing to do with "subjective probabilities" or other merely "epistemic" considerations.

The simplest mathematical formulation that Popper gives of this concept can be found in the tenth chapter of Conjectures and Refutations.. Here he defines it as:

where is the verisimilitude (or truthlikeness) of a, is a measure of the content of truth of a, and is a measure of the content of the falsity of a.

Knowledge, for Popper, was objective, both in the sense that it is objectively true (or truthlike), and also in the sense that knowledge has an ontological status (i.e., knowledge as object) independent of the knowing subject (Objective Knowledge: An Evolutionary Approach, 1972). He proposed three worlds (see Popperian cosmology): World One , being the phenomenal world, or the world of direct experience; World Two , being the world of mind, or mental states, ideas, and perceptions; and World Three, being the body of human knowledge expressed in its manifold forms, or the products of the second world made manifest in the materials of the first world (i.e.–books, papers, paintings, symphonies, and all the products of the human mind). World Three, he argued, was the product of individual human beings in exactly the same sense that an animal path is the product of individual animals, and that, as such, has an existence and evolution independent of any individual knowing subjects. The influence of World Three, in his view, on the individual human mind (World Two) is at least as strong as the influence of World One. In other words, the knowledge held by a given individual mind owes at least as much to the total accumulated wealth of human knowledge, made manifest, as to the world of direct experience. As such, the growth of human knowledge could be said to be a function of the independent evolution of World Three. Compare with Memetics. Contemporary philosophers have not embraced Popper's Three World conjecture, due mostly, it seems, to its resemblance to Cartesian dualism.

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In The Open Society and Its Enemies and The Poverty of Historicism, Popper developed a critique of historicism and a defence of the 'Open Society' and liberal democracy. Historicism is the theory that history develops inexorably and necessarily according to knowable general laws towards a determinate end. Popper argued that this view is the principal theoretical presupposition underpinning most forms of authoritarianism and totalitarianism. He argued that historicism is founded upon mistaken assumptions regarding the nature of scientific law and prediction. Since the growth of human knowledge is a causal factor in the evolution of human history, and since "no society can predict, scientifically, its own future states of knowledge", it follows, he argued, that there can be no predictive science of human history. For Popper, metaphysical and historical indeterminism go hand in hand.

Among his contributions to philosophy is his answer to David Hume's Problem of Induction. Hume stated that just because the sun has risen every day for as long as anyone can remember, doesn't mean that there is any rational reason to believe it will come up tomorrow. There is no rational way to prove that a pattern will continue on just because it has before.

Popper's reply is characteristic, and ties in with his criterion of falsifiability. He states that while there is no way to prove that the sun will come up, we can theorize that it will. If it does not come up, then it will be disproven, but since right now it seems to be consistent with our theory, the theory is not disproven.
Thus, Popper's demarcation between science and non-science serves as an answer to an old logical problem as well. This approach was criticised by Peter Singer for masking the role induction plays in empirical discovery.

By all accounts, Popper has played a vital role in establishing the philosophy of science as a vigorous, autonomous discipline within analytic philosophy, through his own prolific and influential works, and also through his influence on his own contemporaries and students. Popper founded the Philosophy department at the London School of Economics and there lectured and taught both Imre Lakatos and Paul Feyerabend, two of the foremost philosophers of science in the next generation of analytic philosophy. (Lakatos significantly modified Popper's position, and Feyerabend repudiated it entirely, but the work of both is deeply influenced by Popper and engaged with many of the problems that Popper set.)

While there is some dispute as to the matter of influence, Popper had a long standing and close friendship with economist Friedrich Hayek, who was also brought to the London School of Economics from Vienna. Each found support and similarities in each other's work, citing each other often, though not without qualification. In a letter to Hayek in 1944, Popper stated, "I think I have learnt more from you than from any other living thinker, except perhaps Alfred Tarski." (See Hacohen, 2000). Popper dedicated his Conjectures and Refutations to Hayek. For his part, Hayek dedicated a collection of papers, Studies in Philosophy, Politics, and Economics, to Popper, and in 1982 said, "...ever since his Logik der Forschung first came out in 1934, I have been a complete adherent to his general theory of methodology." (See Weimer and Palermo, 1982).

The Quine-Duhem thesis argues that it's impossible to test a single hypothesis on its own, since each one comes as part of an environment of theories. Thus we can only say that the whole package of relevant theories has been collectively falsified, but cannot conclusively say which element of the package must be replaced. An example of this is given by the discovery of the planet Neptune: when the motion of Uranus was found not to match the predictions of Newton's laws, the theory "There are seven planets in the solar system" was rejected, and not Newton's laws themselves. Popper discussed this critique of naïve falsificationism in Chapters 3 & 4 of The Logic of Scientific Discovery. For Popper, theories are accepted or rejected via a sort of 'natural selection'. Theories that say more about the way things appear are to be preferred over those that do not; the more generally applicable a theory is, the greater its value. Thus Newton’s laws, with their wide general application, are to be preferred over the much more specific “the solar system has seven planets”.

Thomas Kuhn’s influential book The Structure of Scientific Revolutions argued that scientists work in a series of paradigms, and found little evidence of scientists actually following a falsificationist methodology. Popper's student Imre Lakatos attempted to reconcile Kuhn’s work with falsificationism by arguing that science progresses by the falsification of research programs rather than the more specific universal statements of naïve falsificationism. Another of Popper’s students Paul Feyerabend ultimately rejected any prescriptive methodology, and argued that the only universal method characterizing scientific progress was anything goes.

Popper seems to have anticipated Kuhn's observations. In his collection Conjectures and Refutations: The Growth of Scientific Knowledge (Harper & Row, 1963), Popper writes, "[S]cience must begin with myths, and with the criticism of myths; neither with the collection of observations, nor with the invention of experiments, but with the critical discussion of myths, and of magical techniques and practices. The scientific tradition is distinguished from the pre-scientific tradition in having two layers. Like the latter, it passes on its theories; but it also passes on a critical attitude towards them. The theories are passed on, not as dogmas, but rather with the challenge to discuss them and improve upon them."

Another objection is that it is not always possible to demonstrate falsehood definitively, especially if one is using statistical criteria to evaluate a null hypothesis. [citation needed] More generally, it is not always clear that if evidence contradicts a hypothesis that this is a sign of flaws in the hypothesis rather than of flaws in the evidence. However, this is a misunderstanding of what Popper's philosophy of science sets out to do. Rather than proffering a set of instructions that merely need to be followed diligently to achieve science, Popper makes clear in The Logic of Scientific Discovery his belief that the resolution of conflicts between hypotheses and observations can only be a matter of the collective judgement of scientists, in each individual case. [4]

Charles Taylor accuses Popper of exploiting his worldwide fame as an epistemologist to diminish the importance of philosophers of the 20th century continental tradition. According to Taylor, Popper's criticisms are completely baseless, but they are received with an attention and respect that Popper's "intrinsic worth hardly merits". [6]

In 2004philosopher and psychologistMichel ter Hark (Groningen, The Netherlands) published a book, called Popper, Otto Selz and the rise of evolutionary epistemology, in which he claims that Popper got part of his ideas from his tutor, the German-Jewish psychologist Otto Selz. Selz himself never published his ideas, partly because of the rise of Nazism which forced him to quit his work in 1933, and the prohibition of referencing to Selz' work.

The Lesson of this Century, Interviewer: Giancarlo Bosetti, English translation: Patrick Camiller), 1992

The World of Parmenides, Essays on the Presocratic Enlightenment, (Edited by Arne F. Petersen with the assistance of Jørgen Mejer)

The Myth of the Framework: In Defence of Science and Rationality, 1994

Knowledge and the Mind-Body Problem: In Defence of Interactionism, 1994

Quantum Theory and the Schism in Physics, (From the Postscript to The Logic of Scientific Discovery)

In Search of a Better World, a collection of Popper’s essays and lectures covering a range of subjects from the beginning of scientific speculation in classical Greece to the need for a new professional ethic based on the ideas of tolerance and intellectual responsibility; "All things living are in search of a better world."; Karl Popper, from the Preface of the book.

Edmonds, D., Eidinow, J. Wittgenstein's Poker. New York: Ecco 2001. A review of the origin of the conflict between Popper and Ludwig Wittgenstein, focused on events leading up to their volatile first encounter at 1946 Cambridge meeting.

Feyerabend, Paul Against Method. London: New Left Books, 1975. A polemical, iconoclastic book by a former colleague of Popper's. Vigorously critical of Popper's rationalist view of science.

Kadvany, John Imre Lakatos and the Guises of Reason. Durham and London: Duke University Press, 2001. ISBN 0-8223-2659-0. Explains how Imre Lakatos developed Popper's philosophy into a historicist and critical theory of scientific method.

Kuhn, Thomas S. The Structure of Scientific Revolutions. Chicago: University of Chicago Press, 1962. Central to contemporary philosophy of science is the debate between the followers of Kuhn and Popper on the nature of scientific enquiry. This is the book in which Kuhn's views received their classical statement.

Levinson, Paul, ed. In Pursuit of Truth: Essays on the Philosophy of Karl Popper on the Occasion of his 80th Birthday. Atlantic Highlands, NJ: Humanities Press, 1982. A collection of essays on Popper's thought and legacy by a wide range of his followers. Includes an interview with Sir Ernst Gombrich.

Magee, Bryan Popper. London: Fontana, 1977. An elegant introductory text. Very readable, albeit rather uncritical of its subject, by a former Member of Parliament.

Magee, Bryan Confessions of a Philosopher, Weidenfeld and Nicolson, 1997. Magee's philosophical autobiography, with a chapter on his relations with Popper. More critical of Popper than in the previous reference.

Munz, Peter Beyond Wittgenstein's Poker: New Light on Popper and Wittgenstein Aldershot, Hampshire, UK: Ashgate, 2004. ISBN 0754640167. Written by the only living student of both Wittgenstein and Popper, an eyewitness to the famous "poker" incident described above (Edmunds & Eidinow). Attempts to synthesize and reconcile the differences between these two philosophers.

Schilpp, Paul A., ed. The Philosophy of Karl Popper, 2 vols. La Salle, IL: Open Court Press, 1974. One of the better contributions to the Library of Living Philosophers series. Contains Popper's intellectual autobiography, a comprehensive range of critical essays, and Popper's responses to them.

Stokes, G. Popper: Philosophy, Politics and Scientific Method. Cambridge: Polity Press, 1998. A very comprehensive, balanced study, which focuses largely on the social and political side of Popper's thought.